ECE 222b Applied Electromagnetics Notes Set 5

Transcription

1 ECE b Applied Electomagnetics Notes Set 5 Instucto: Pof. Vitaliy Lomakin Depatment of Electical and Compute Engineeing Univesity of Califonia, San Diego 1

2 Auxiliay Potential Functions (1)

3 Auxiliay Potential Functions () 3

4 Auxiliay Potential Functions (3) To uniquely detemine A: A = 0 Coulomb gauge condition Vecto Poisson s equation 4

5 Auxiliay Potential Functions (4) 5

6 Auxiliay Potential Functions (5) 6

7 Auxiliay Potential Functions (6) Loenz gauge condition 7

8 Auxiliay Potential Functions (7) Conside the case of magnetic souce : D = 0 D = F m m E m 1 = F ε 8

9 Auxiliay Potential Functions (8) Loenz gauge condition 9

10 Auxiliay Potential Functions (9) Summay: 10

11 Auxiliay Potential Functions (10) Diect appoach: Compae two appoaches. 11

12 Solution of Vecto Potential (1) Vecto wave equation Scala wave equations Conside Let 1

13 Solution of Vecto Potential () Pinciple of linea supeposition: Let Fo : 13

14 Solution of Vecto Potential (3) In spheical coodinates: 1 G sin G G G G = + θ + = sinθ θ θ 1 sinθ ϕ G G G 1 + kg= kg= Since ( G) = G ( G) G G 1 = ( ) G 1 k G 1 1 G + ( ) = 0 14

15 Solution of Vecto Potential (4) Two solutions Hence, To detemine C, G = Ce G = Ce jk 1 1 jk 1 1 non-physical physical e G = C jk

16 Solution of Vecto Potential (5) Since G = G S ε Gds + k Gdv = V ε 1 4 πc 0 C = 1 4π Theefoe, 16

17 Solution of Vecto Potential (6) 17

18 Solution of Vecto Potential (7) Summay: Fo suface cuents: Fo line cuents: 18

19 Dipole Radiation (1) Example #1: z In spheical coodinate system: x µ Il jk A = Az cosθ = e cosθ 4π µ Il jk Aθ = Az sinθ = e sinθ 4π A = 0 ϕ I l l 0 electic dipole y 19

20 Dipole Radiation () 0

21 Dipole Radiation (3) E E E θ ϕ Il cosθ 1 = η 1 + π jk e jk jkil sinθ 1 1 = η 1+ 4 π jk ( k) = 0 e jk Fo >> λ : (fa-field zone) jkil sinθ jk Eθ = η e = ηh 4π jkil sinθ jk Eθ Hϕ = e = 4π η ϕ Fa field 1

22 Dipole Radiation (4) Time aveage adiated powe

23 E Dipole Radiation (5) E decays apidly E θ and H φ dominate E θ and H φ popagate in phase No adiation along the z-axis 3

24 Dipole Radiation (6) Example #: L z R θ y x L Conside >> z : = + R z z cosθ 1/ z cosθ z cosθ R 1 = 1 z cosθ 4

25 Dipole Radiation (7) Conside L jkz cosθ I = k z e dz L L sin L 0 L jkz cosθ L jkz cosθ = sin k z e dz sin k z e dz L 5

26 Dipole Radiation (8) L L L jkz cosθ L jkz cosθ = sin k z e dz sin k z e dz L L = sin 0 k z cos( cos ) kz θ dz L L = sin k z d sin( kz cos θ ) k cosθ 0 L sin L L sin( cos ) cos L sin( cos ) = k z kz θ + k k z kz θ dz k cosθ 0 0 L L = cos k z sin( kz cos θ ) dz cosθ 0 6

27 Dipole Radiation (9) L L = cos k z d cos( kz cos θ ) k cos θ 0 L = cos k z cos( kz cos θ ) k cos θ L L k sin k z cos( kz cos θ ) dz 0 L L 1 = cos k cosθ cos k I k cos θ + cos θ L 0 I = L L cos k cosθ cos k k sin θ 7

28 Dipole Radiation (10) Vecto potential: A z L L cos k cosθ cos k µ I m jk = e 4π k sin θ Fa field: E θ H ϕ = = L L cos k cosθ cos k jk jη Ime π sinθ Eθ η 8

29 Dipole Radiation (11) L = λ 4 L = λ L = λ L = 3 λ 9

30 30 Fa-Field Appoximation (1) dv R e M F dv R e J A jkr V jkr V = = ) ( 4 ) ( 4 π ε π µ cosψ cosψ R + = In the fa-field zone: ( ) N e dv Je e dv e J A jk jk V jk jk V = = = π µ π µ π µ ψ ψ cos cos dv Je N jk V = cosψ

31 Fa-Field Appoximation () E 0 E jω A θ θ E jω A ϕ ϕ H H H θ ϕ 0 Eϕ = η Eθ = η jω A η ϕ jω A η θ 31

32 Fa-Field Appoximation (3) H 0 E 0 H jω F E ηh = jωηf H jω F E ηh = jωηf θ θ θ ϕ ϕ ϕ ϕ ϕ θ θ Total fields: E 0 E jωa jωη F = jω( A + ηf ) θ θ ϕ θ ϕ E jω A + jωη F = jω( A ηf ) ϕ ϕ θ ϕ θ 3

33 Fa-Field Appoximation (4) H 0 jω jω Hθ Aϕ jωfθ = ( Aϕ ηfθ) η η jω jω Hϕ Aθ jωfϕ = ( Aθ + ηfϕ) η η o E 0 jk jke Eθ ( Lϕ + ηnθ) 4π jk jke Eϕ ( Lθ ηnϕ) 4π H H H θ ϕ 0 jke 4π jk N ϕ jk jke Nθ + 4π Lθ η Lϕ η 33

34 Fa-Field Appoximation (5) Sommefeld adiation condition (Descibes the field behavio fa away fom the souce---can be egaded as the bounday condition at the infinity) 34

35 Fa-Field Appoximation (6) Example: x I z ψ R y 35

36 Fa-Field Appoximation (7) N = N sinϕ + N ϕ 0 0 x π 0 y cosϕ jka sinθ cos( ϕ ϕ') = Ia cos( ϕ ϕ') e dϕ' π ϕ jka sinθ cos Φ = Ia cos Φ e dφ ϕ π jka sinθ cos Φ π jka sinθ cos Φ 0 cos 0 0 cos π = Ia Φ e dφ + Ia Φ e dφ π jka sinθ cos Φ 0 cos 0 0 = Ia Φ e dφ Ia cos Φ e = I aπ jj ( ka sin θ) I aπ jj ( ka sin θ) jka sinθ cos Φ n ( ) ( ) = πji aj ( ka sin θ) J z = ( 1) J z π n n dφ N = N sinθ cosϕ + N sinθsinϕ = 0 x y N = N cosθ cosϕ + N cosθsinϕ = 0 θ x y 36

37 Fa-Field Appoximation (8) E E θ 0 0 aωµ I 0 jk Eϕ e J1 ka ( sin θ ) H H H θ ϕ 0 jke Nϕ = 4π 0 jk E ϕ η Assume a << λ, J ( ka sin θ) E H ϕ θ = η Eϕ = η 1 ( ka) I sin 0 4 θ e jk ka sinθ 37

38 Fa-Field Appoximation (9) The field due to a magnetic dipole E ϕ = jkkl sinθ e 4π Kl jk is Theefoe, if Kl = jsωµ I 0 the magnetic dipole is equivalent to an electic cuent loop. 38

39 Field-Souce Relations (1) Souce Vecto potentials Fields 39

40 Field-Souce Relations () Define then 40

41 Field-Souce Relations (3) Electic dyadic Geen s function Magnetic dyadic Geen s function 41

42 Field-Souce Relations (4) Dyadic Geen s functions: 4